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Let R be a commutative ring with unity and let G be a group. The group ring RG is a free R-module having the elements of G as a basis, with multiplication induced byThe first theorem in this paper deals with idempotents in RG and improves a result of Connell. In the second section we consider the Jacobson radical of RG, and we prove a theorem about a class of algebras that includes RG when G is locally finite and R is an algebraically closed field of characteristic zero. The last theorem shows that if R is a field and G is a finite nilpotent group, then RG determines RP for every Sylow subgroup P of G, regardless of the characteristic of R.

Several concepts in discrete mathematics such as block designs, Latin squares, Hadamard matrices, tactical configurations, errorcorrecting codes, geometric configurations, finite groups, and graphs are by no means independent. Combinations of these notions may serve the development of any one of them, and sometimes reveal hidden interrelations. In the present paper a central role in this respect is played by the notion of strongly regular graph, the definition of which is recalled below.In § 2, a fibre-type construction for graphs is given which, applied to block designs withλ= 1 and Hadamard matrices, yields strongly regular graphs. The method, although still limited in its applications, may serve further developments. In § 3 we deal with block designs, first considered by Shrikhande[22],in which the number of points in the intersection of any pair of blocks attains only two values.

This paper continues a study appearing in (5) of the combinatorial properties of a matrix A of m rows and n columns, all of whose entries are 0's and 1's. Let the sum of row i of A be denoted by r i and let the sum of column i of A be noted by st. We call R = (r 1, … , rm) the row sum vector and S = (s 1, … , s n) the column sum vector of A. The vectors R and S determine a class consisting of all (0, 1)-matrices of m rows and n columns, with row sum vector R and column sum vector S. Simple arithmetic properties of R and S are necessary and sufficient for the existence of a class (1 ; 5).

This paper is a result of an investigation into general methods of determining the irreducible characters of GL(n, q), the group of all non-singular linear substitutions with marks in GF(q), and of the related groups, SL(n, q), PGL(n, q), PSL(n, q), the corresponding group of determinant unity, projective group, projective group of determinant unity, respectively.

Corresponding to every graph, bipartite graph, or directed bipartite graph there exists a directed graph which is connected if and only if the original graph is connected.In this paper, it is shown that for every directed graph there exists a certain bipartite graph such that the directed graph is connected if and only if the bipartite graph is irreducible. Other connections between reducibility and connectivity are established.

In 1927 J. H. Redfield (9) stressed the intimate interrelationship between the theory of finite groups and combinatorial analysis. With this in mind we consider Pólya's theorem (7) and the Redfield-Read superposition theorem (8, 9) in the context of the theory of permutation representations of finite groups. We show in particular how the Redfield-Read superposition theorem can be deduced as a special case from a simple extension of Pólya's theorem. We give also a generalization of the superposition theorem expressed as the multiple scalar product of certain group characters. In a later paper we shall give some applications of this generalization.

1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form(1) where . It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2k) = 1, this representation being the number 2k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral

The present paper is a sequel to that of J. H. Chung (2) and contains a proof of a conjecture made by him, namely, that the number of ordinary (modular) irreducible representations contained in a given p-block of Sn is independent of the p-core. A summary of the results contained herein appeared in the Proceedings of the National Academy of Sciences (9).

The purpose of this paper is to investigate non-linear recursive sequences of maximum length with elements from GF(2). In particular, the question of whether or not a recursive sequence of maximum length can be equal to its dual is settled. This question, as far as the author knows, was originally asked by Rosser. Part I contains the necessary background for Part II, and in the main is a condensation of some unpublished work (1955) of W. A. Blankinship and R. P. Dilworth.

A finite net N of degree k, order n, is a geometrical object of which the precise definition will be given in §1. The geometrical language of the paper proves convenient, but other terminologies are perhaps more familiar. A finite affine (or Euclidean) plane with n points on each line is simply a net of degree n+ 1, order n (Marshall Hall [1]). A loop of order n is essentially a net of degree 3, order n (Baer [1], Bates [1]). More generally, for , a set of k —2 mutually orthogonal n ⨯ n latin squares may be used to define a net of degree k, order n (and conversely) by paralleling Bose's correspondence (Bose [1]) between affine planes and complete sets of orthogonal latin squares.

In our paper “Magic graphs” (1) we showed that every complete graph Kn with n ⩾ 5 is “magic,” i.e., if the vertex set is indicated {vi} and if eij is the edge joining vi and vj, i ≠ j , then there exists a function α(eij) such that the set {α(eij)} consists of distinct positive rational integers and the vertex sums1have a constant value σ(α) for k = 1, 2, … , n. We noted that K2 is magic and showed that K3 and K4 are not magic.

Let G be a finite group and A a group of automorphisms of G. Clearly A acts as a permutation group on G#, the set of non-identity elements of G. We assume that this permutation representation is half transitive, that is all the orbits have the same size. A special case of this occurs when A acts fixed point free on G. In this paper we study the remaining or non-fixed point free cases. We show first that G must be an elementary abelian g-group for some prime q and that A acts irreducibly on G. Then we classify all such occurrences in which A is a p-group.

Anhbyhmatrix with entries ±1 is called amodular Hadamard matrixif the inner product of any two distinct row vectors is a multiple of a fixed (positive) integern;such a matrix is also referred to as an“H(n, h) matrix” with parametersnandh.Modular Hadamard matrices and the related combinatorial designs were introduced in [2]; that paper was concerned mainly with two of the related designs, the “pseudo (ν, k, λ)- designs” and the “ (m, v, k1,λ1,k2,λ2,f, λ3)-designs” (the reader is referred to [2] for the definition of these designs).

A pseudo-complemented lattice is a lattice L with zero such that for every a ∊ L there exists a* ∊ L such that, for all x ∊ L, a ∧ x = 0 if and only if x ≦ a*. a* is called a pseudo-complement of a. It is clear that for each element a of a pseudo-complemented lattice L, a* is uniquely determined by a. Thus * can be regarded as a unary operation on L. Moreover, each pseudo-complemented lattice contains the unit, namely 0*. It follows that every pseudo-complemented lattice L can be regarded as an algebra (L; (∧, ∨, *, 0, 1)) of type (2, 2, 1, 0, 0). In this paper, we consider only distributive pseudo-complemented lattices. For simplicity, we call such a lattice a p-algebra.

1. In 1948 Heilbronn [4] proved the following theorem.THEOREM H. For every real θ and every positive integer N, there is an integer n satisfying(1.1)whereis an arbitrarily small number,depends only on, and ‖t‖ means the distance from t to the nearest integer.The interest of the result (1.1) is that the inequality is uniform in θ, and is therefore analogous to the classical inequality of Dirichlet for the fractional part of θn. In this paper we shall prove the following theorem.THEOREM. For every real θ and every positive integer N, there is an integer n satisfying(1.2)where A is an absolute constant and.

Let a1, a2, … , at be a set of groupwise relatively prime positive integers. Several authors, (2; 3; 5; 6), have determined bounds for the function F(a1, …, at) defined by the property that the equation1has a solution in positive integers X1, …, xt for n > F(a1, ..., at). If F(a1, …, at) is a function of this type, it is easy to see that2is the corresponding function for the solvability of (1) in non-negative x's.It is well known that a1a2 is the best bound for F(a1, a2) and a1a2 — a,1 — a2 for G(a1, a2). Otherwise only in very special cases have the best bounds been found, even for t = 3.In the present paper a symmetric expression is developed for the best bound for F(a1, a2, a3) which solves that problem and gives insight on the general problem for larger values of t. In addition, some relations are developed which may be of interest in themselves.

In this paper we study the following problem originally proposed by Walsh (8). To determine the class of functions f(x) continuous in a given n-dimensional region R and having the property that the value of f(x) be equal to the average of f(x) over the vertices of all sufficiently small regular polytopes similar to a given one, which are centred at x. This problem has been studied by several mathematicians (1; 6; 8) and has been completely solved except for the four-dimensional regular polytopes {3, 4, 3}, {3, 3, 5}, {5, 3, 3} (see 3, p. 129, for the meaning of these symbols) and the n-dimensional cube. In each case, the class of functions is identical with a class of harmonic polynomials which can be specified. In § 2, we solve the problem for the four-dimensional figures, thus leaving the problem open only for the n-dimensional cube.

In [2], we considered a class of Lie algebras generalizing the classical simple Lie algebras. Using a field Φ of characteristic zero and a square matrix (Aij) of integers with the properties (1) Aii = 2, (2) Aij ≦ 0 if i ≠ j, (3) Aij = 0 if and only if Ajt = 0, and (4) is symmetric for some appropriate non-zero rational a Lie algebra E = E((Aij)) over Φ can be constructed, together with the usual accoutrements: a root system, invariant bilinear form, and Weyl group.For indecomposable (A ij), E is simple except when (Aij) is singular and removal of any row and corresponding column of (Aij) leaves a Cartan matrix. The non-simple Es, Euclidean Lie algebras, were our object of study in [3] as well as in the present paper. They are infinite-dimensional, have ascending chain condition on ideals, and proper ideals are of finite codimension.

1. For an entire function , let M(r, f), μ(r, f), and v(r, f) denote the maximum modulus, the maximum term, and the rank for |z\ = r, respectively. Also, letand λ(r) the lower proximate order relative to log M(r, f). For the properties of the lower proximate order, we refer the reader to the paper by Shah (1).2. We prove the following theorems.THEOREM 1. For an entire functionwhere μ(r, f1) and M(r, f1) correspond to fl(z), the derivative of f(z), provided (n + l)Rn > nRn+1, when L(f) > 1.

In this paper we examine the classGof simple undirected, connected graphs of diameterd> 1, girth2d,and for anygɛG, if a pair of nodes are at distancedfrom each other, then that pair of nodes is connected bytdistinct paths of lengthd, t> 1. (The girth ofgis the length of the smallest circuit ing.)We establish, in § 2, that for allgɛG, gis regular.We establish necessary conditions for the existence of elements ofG.IfgɛG,we adopt the notationg=g(d, t, v, n),wherevis the valence ofgandnis the number of nodes. It is of course possible forg, hɛG, g ≠ h,and for givend, t, v, nto have bothg(d, t, v, n)andh(d, t, v, n).In particular, we show that ifd= 2,t≠ 2, 4 or 6, then there is at most a finite number of graphs with a particular giventvalue.

A method of proving the impossibility of certain Affine Resolvable Balanced Incomplete Block Designs (A.R.B.I.B.D.) has been given by the author elsewhere [9]. More complete results in the same direction are obtained here using the ideas of a paper by Connor [4].

Let G be a compact group. According to the celebrated theorem of Peter-Weyl there exists a complete set of finite-dimensional irreducible unitary representations of G, the completeness meaning that for any group element other than the identity there is a representation sending it into a matrix other than the unit matrix. If G is commutative, the representations are necessarily one-dimensional. It is an immediate consequence of the Peter-Weyl theorem that the converse also holds: if every representation is one-dimensional, G is commutative. The main theorem in the present paper is a generalization of this result to the case where the representations have bounded degree.

If a group contains two subgroups A and B such that every element of the group is either in A or can be represented uniquely in the form aba', a, a’ in A, b ≠ 1 in B, we shall call the group an independent ABA-group. In this paper we shall investigate the structure of independent ABA -groups of finite order.A simple example of such a group is the group G of one-dimensional affine transformations over a finite field K. In fact, if we denote by a the transformation x’ = ωx, where ω is a primitive element of K, and by b the transformation x’ = —x + 1, it is easy to see that G is an independent ABA -group with respect to the cyclic subgroups A, B generated by a and b respectively.Since G admits a faithful representation on m letters (m = number of elements in K) as a transitive permutation group in which no permutation other than the identity leaves two letters fixed, and in which there is at least one permutation leaving exactly one letter fixed, G is an example of a Frobenius group.

In an earlier paper [3] I studied the property С and related properties С′ and С; but the principal problem, viz, to prove, with the axiom of choice only (without any other hypothesis), the existence of a nondenumerable set of property С, remains open.

In this paper all representations are over the complex field K. The generalized symmetric group S(n, m) of order n!mn is isomorphic to the semi-direct product of the group of n × n diagonal matrices whose rath powers are the unit matrix by the group of all n × n permutation matrices over K. As a permutation group, S(n, m) consists of all permutations of the mn symbols {1, 2, …, mn} which commute withObviously, S (1, m) is a cyclic group of order m, while S(n, 1) is the symmetric group of order n!. If ci = (i, n+ i, …, (m – 1)n+ i) andthen {c1, c2, …, cn} generate a normal subgroup Q(n) of order mn and {s1, s2, …, sn…1} generate a subgroup S(n) isomorphic to S(n, 1).

Let C be a square matrix with complex elements. If C = C' (C' denotes the transpose of C) there exists a unitary matrix U such that(1)where the μ's are the non-negative square roots of the eigenvalues μ12, μ22, … , μn2 of C*C (C* is the adjoint of C) (2). If C is skew-symmetric, that is, C= — C”, there exists a unitary matrix U such that(2)where(3)and the α's are the positive square roots of the non-zero eigenvalues α12, α22, … , αk2 of C*C (1). Clearly rank C = 2k and the number of zeros appearing in (2) is n — 2k. Both (1) and (2) are classical. In a recent paper (3) Stander and Wiegman, apparently unaware of (1), give an alternative derivation of (2) with its appropriate generalization to quaternions.

1. Introduction. Let Q be a matrix of order v, all of whose entries are 0's and l's. Let the total number of l's in Q be t, and let the absolute value of the determinant of Q be denoted by |det Q|. In this paper we study the problem of determining the maximum of |det Q| for fixed t and v. It turns out that this problem is closely related to the v, k, λ problem, which has been extensively studied of late.

The kth rank number, rankkB, of a differentiable arc B in real projective n-space is the least upper bound of the number of osculating k-spaces of B which meet an (n – k – l)-flat, k = 0, 1, …, n – 1. The number rank0B is called the order of B; cf. 1.1-1.3. It has been conjectured by Peter Scherk that(0.1)equality holding if and only if B has the order n; cf. [2, p. 396]. In this paper we prove the following results.THEOREM 1. If B is a differentiable elementary arc, then (0.1) holds for k = 0, 1, …, n – 1.THEOREM 2. If B is a differentiable elementary arc and order B > n, then rankkB > (k + 1) (n – k) for k = 1, …, n – 2.By a theorem of Park [3, p. 38], every differentiable arc contains a subarc of order n. This eliminates the assumption that B is elementary from Theorem 1. We do not know whether it can be dropped from Theorem 2.

1. Introduction. This paper is concerned with problems connected with the permutations of the integers 1, 2, … , n subject to certain special restrictions. One such class of problems, the so-called “card matching” problems, deals with conditions of the type, “the number i is in the jth position,” “the number k is in the mth position,” etc.

We define below a notion for modules which is dual to that of faithful, and a notion of “fully divisible” which generalizes that of injectivity. We show that the bicommutator of a cofaithful, fully divisible left R-module is isomorphic to a subring of Qmax(R), the complete ring of left quotients of R.In recent papers, Goldman [2] and Lambek [3] investigated rings of left quotients of a ring R constructed with respect to torsion radicals. It is known that every ring of left quotients of R is isomorphic to the bicommutator of an appropriate injective left R-module. We investigate below subrings of rings of quotients which are determined by radicals rather than torsion radicals, and show that any such ring can be constructed as the bicommutator of a fully divisible left R-module.

This paper contains an account of a simple method of deriving the coordinates of the vertices of any uniform polytope or honeycomb (degenerate polytope) whose symmetry group is generated by reflections.

It will be shown in this paper that if two chords of a closed plane convex curve θ divide θ into four arcs of equal length and intersect inside the domain bounded by θ, then the sum of the lengths of the two chords is at least equal to (√5 — 2)½ times the length of θ. We shall show firstly that we need only consider the case when θ is a convex (possibly degenerate) quadrilateral and then prove the result in this case.This result is related to a conjecture of P. Ungar in which the chords are assumed to be perpendicular and the factor (√5 — 2)½ is replaced by §. But Ungar's conjecture is neither proved nor disproved by this result.

F. B. Jones (6) has shown that, if , then every separable normal Moore space is metrizable. It is not known whether this assumption is necessary, though perhaps some progress is made in (5). However, it is easily seen from R. H. Bing's Example E in (3) that a certain condition (see (1) below) implied by is necessary. Also in (3), Bing showed that every screenable normal Moore space is metrizable. In this paper we establish that: (1) every separable normal Moore space is metrizable if and only if every uncountable subspace M of E1 contains a subset which is not an Fσ (in M); (2) if every pointwise paracompact normal Moore space is metrizable, then so is every separable normal Moore space; (3) every screenable Moore space is pointwise paracompact but not conversely; (4) a T3-space is a pointwise paracompact Moore space if and only if it has a uniform base (in the sense of (1, p. 40), not a uniformity).

In 1967 Foulser [1] defined a class of translation planes, called generalized André planes or λ-planes and discussed the associated autotopism collineation groups. While discussing these collineation groups he raised the following question:“Are there collineations of a λ plane which move the axes but do not interchange them?”.In this context, Foulser mentioned a conjecture of D. R. Hughes that among the André planes, only the Hall planes have collineations moving the axes without interchanging them. Wilke [4] answered Foulser's question partially by showing that the conjecture of Hughes is indeed correct. Recently, Foulser [2] has shown that possibly with a certain exception the Hall planes are the only generalized André planes which have collineations moving the axes without interchanging them. Our aim in this paper is to give an alternate proof, which is completely general, and is in the style of the original problem.

It is the purpose of this paper to show that many of the enumerative techniques available for counting rooted plane trees may be extended to tree-rooted maps, that is, rooted maps in which a spanning tree is distinguished as root tree. For example, tree-rooted maps are enumerated by partition, and the average number of trees in a rooted map with n edges is determined. An enumerative similarity between Hamiltonian rooted maps (that is, rooted maps with a distinguished Hamiltonian polygon) and tree-rooted maps is discussed. A 1-1 correspondence is established between treerooted maps with n edges and Hamiltonian rooted trivalent maps with 2n + 1 vertices in which the root vertex is exceptional, being divalent, both of which are in 1-1 correspondence with non-separable Hamiltonian-rooted triangularized digons with n internal vertices, where both the latter are as defined in (2).

The generalized Lototsky transform (or the [F, dn] transform) of a sequence {sn} into a sequence {tn) was defined in (2) in the following way: Let {dn) (dn ≠ —1) be a fixed complex sequence and letthen the sequence {tn} is defined byIn a recent paper by V. F. Cowling and C. L. Miracle (1) on these transformations, the following two problems were left open.

The purpose of this paper is to prove the following theorem:Theorem. Given a convex curve C, of perimeter length I and three points M, N, and P which divide the perimeter of C into three parts of equal length, the perimeter length of the triangle MNP is never less than ½l. Equality holds only in the case where C is an equilateral triangle and M, N, and P are the mid-points of the three sides.

LetMbe a smooth, closed, simply connected manifold of dimension greater than 5. LetTbe an involution onMwith a positive, finite number of fixed points. Our aim in this paper is to prove the following theorem (which is somewhat like that of Wasserman (7)).

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

For a distributive lattice L, let denote the poset of all prime ideals of L together with ∅ and L. This paper is concerned with the following type of problem. Given a class of distributive lattices, characterize all posets P for which for some . Such a poset P will be called representable over. For example, if is the class of all relatively complemented distributive lattices, then P is representable over if and only if P is a totally unordered poset with 0, 1 adjoined. One of our main results is a complete characterization of those posets P which are representable over the class of distributive lattices which are generated by their meet irreducible elements. The problem of determining which posets P are representable over the class of all distributive lattices appears to be very difficult.

A proper subgraph of a connected linear graph is said to disconnect the graph if removing it leaves a disconnected graph. In this paper we characterize, in the following sense, the disconnecting subgraphs of a fixed connected graph. We define two distinct types of disconnecting subgraphs (isthmuses and articulators) which are minimal in the sense that no proper subgraph of either type can disconnect the graph. We then show that any disconnecting subgraph must contain either an isthmus or an articulator. We also define a set of subgraphs (called dense) which form a lattice. We show that the union of the minimal dense subgraphs contains all isthmuses and articulators. In terms of these subgraphs we investigate some of the consequences of assuming that a disconnecting subgraph must contain at least m points.

In a recent paper (1) the fifty-eight metabelian groups of order p11 that are generated by five elements and have all their elements of order p were determined and characterized in terms independent of any particular selection of the generating elements. In dealing with fifty-seven of these groups there was no occasion to distinguish between one odd prime and another, except that in exhibiting canonical forms it was necessary to select irreducible polynomials and these, of course, depended on p. The fifty-eighth group was described in two ways in terms that were independent of p, but the proof of uniqueness could not be made without taking into account properties of p. These properties distribute the primes into classes, and the properties are reflected in the groups of order p11 in characteristic subgroups some of which exist for one prime and not for another. It may be that examination of the groups of isomorphisms of some of the fifty-seven groups would produce characteristic subgroups for one p that would not exist for another, but the writer considers it doubtful.

In earlier papers (4; 5; 6), certain space curves, invariant under cyclic involutions of periods three, five, and seven, have been mapped. Lucien Godeaux (2; 3) in 1916 mapped plane cubic curves, invariant under an involution of period three, onto a cubic surface in ordinary three-space. Mlle. J. Dessart (1) in 1931 mapped plane quintic curves, invariant under an involution of order five, onto a fifth order surface in a space of four dimensions.

Banach (1, pp. 242-243) defines, for two Banach spaces X and Y, a number (X, Y) = inf (log (‖L‖ ‖L-1‖)), where the infimum is taken over all isomorphisms L of X onto F. He says that the spaces X and Y are nearly isometric if (X, Y) = 0 and asks whether the concepts of near isometry and isometry are the same; in particular, whether the spaces c and c0, which are not isometric, are nearly isometric. In a recent paper (2) Michael Cambern shows not only that c and c0 are not nearly isometric but obtains the elegant result that for the class of Banach spaces of continuous functions vanishing at infinity on a first countable locally compact Hausdorff space, the notions of isometry and near isometry coincide.

In this paper we obtain a partial answer in graph-theoretic form to a question raised by Ryser (2, p. 68) concerning the minimal number of interchanges required to transform equivalent (0, l)-matrices into each other.For given positive integersmandnwe consider the collection ofm×n(0, 1)-matricesA= {aij}, i.e.aij= 0 or 1 for 1 ≤i≤m, 1 ≤j≤n. We say the (0, 1)-matricesA= {aij} andB= {bij} areequivalentand writeA~Bif and only if they have the same row and column sums, that is, if and only if.

In this paper we consider finite graphs without loops and multiple edges. A graph is considered to be an ordered pair 〈G, *〉 where G is a finite set the elements of which are called the vertices of while * is a subset of [G]2 (where [G]2 is the set of all subsets of two elements of G). The elements of * are called the edges of . If {P, Q} ∊ *, we say that Q is adjacent to P. The degree of a vertex is the number of vertices adjacent to it. Let k be an integer. We say that is the complete k-graph if G has k elements and * = [G]2.

All the matrices considered in this paper have their elements in the field of residues mod 2.Two non-singular matrices are equivalent if each row of either matrix is a linear combination of rows of the other. The matrices then have equal numbers of rows and equal numbers of columns.A nodal matrix is a non-singular matrix in which no column has more than two l's. A graphic matrix is a non-singular matrix equivalent to a nodal matrix.

SummaryIn this paper we derive a functional equation whose solution would give the sum of the chromatic polynomialP(M,λ) over certain classes of rooted planar mapsMcalled “ triangulations” and “near-triangulations”. For an integral colour-number λ this sum is the number of λ-coloured rooted maps of the kind considered, but the sum can also be discussed for non-integral λ.

In this paper finite groups with the property M, that every maximal subgroup has prime or prime square index, are investigated. A short but ingenious argument was given by P. Hall which showed that such groups are solvable.B. Huppert showed that a finite group with the property M, that every maximal subgroup has prime index, is supersolvable, i.e. the chief factors are of prime order. We prove here, as a corollary of a more precise result, that if G has property M and is of odd order, then the chief factors of G are of prime or prime square order. The even-order case is different. For every odd prime p and positive integer m we shall construct a group of order 2apb with property M which has a chief factor of order larger than m.

It has been shown elsewhere (1, 4) that the number of rooted non-separable planar maps with n edges isIn the present paper we improve upon this result by finding the number fi,j of rooted non-separable planar maps with i + 1 vertices and j + 1 faces. We use the definitions of (1).Among the non-separable planar maps only the loop-map and the link-map have i = 0 or j = 0. We therefore confine our attention to the case in which i and j are both positive.